Any angle can be assigned a rational number by dividing a standard angle, say 90deg, into an arbitrary number of segments geometrically, just as points on the line can.

Then any angle can be defined (given a number) by a geometric cut in precisely the same way that any point on the line can.

There is so much wrong with this statement. First of all, the set of rational numbers is not continuous, which means that it would be impossible to define the sine function as a continuous function.

Also, not all points on the line can be expressed as a ratio of two known lengths. The standard counterexample is the diagonals of a unit square, of length $\displaystyle \begin{align*} \sqrt{2} \end{align*}$.

Any angle in radians is pi times the angle in degrees,

No it's not. Actually any angle in radians is $\displaystyle \begin{align*} \frac{\pi}{180} \end{align*}$ times the angle in degrees. Surely since the radian measure is defined as the number of lengths of the radius swept out on the circumference, and the circumference is equal to $\displaystyle \begin{align*} 2\pi \cdot \,\textrm{radius} \end{align*}$, there are $\displaystyle \begin{align*} 2\pi \end{align*}$ radians in a circle, and thus the ratio of degrees to radians is:

$\displaystyle \begin{align*} 360^{\circ} &: 2\pi ^{C} \\ 1^{\circ} &: \frac{2\pi^C}{360} \\ 1^{\circ} &: \frac{\pi^C}{180} \end{align*}$

The rational geometric definirion of angles by rational numbers (degrees), which is more fundamental than radians, was done by the Sumerians 5000 years ago.

Which was before the time when the Pythagoreans realised the existence of irrational numbers.

I also have a problem with you saying degrees are more fundamental than radians. Radian measurement is the fundamental measurement, it being a continuous DIMENSIONLESS amount, making it logical to substitute into functions (after all, a dimensionless quantity is just a number). Also, nearly all mathematics involving trigonometric functions involves noting some sort of oscillation, which can only happen when you are allowed to continually go around the circle in both directions as many times as necessary, as opposed to degrees, which by definition ARE a ratio, and so making any "negative angle" or "angle greater than $\displaystyle \begin{align*} 360^{\circ} \end{align*}$" meaningless.

With any angle defined geometrically for all x, trigonometric functions are defined geometrically by the standard definitions for all x, and can be treated analytically without series.

I agree with this statement. And the radian measure is defined geometrically as the number of lengths of the radius on the circumference.

And note that __now__ the standard geometric proof of sinx/x, using geometry and the squeeze theorem, is valid.

As it always has been without your nit-picking...

I note in passing that with x and sinx defined analytically without series, SlipEternal’s proof of (x-sinx)/x

^{3} < 16, in

http://mathhelpforum.com/calculus/229484-challenging-limit.html , post#7, which I am locked out of, is correct, but if I were a teacher I wouldn’t give him credit because he didn’t define sinx analytically without series, which in all fairness, he couldn’t have done without this post.

I have a real issue with this statement. In mathematics, it is expected that things are "well known", such as the geometric definitions of the trigonometric functions. It is IMPOSSIBLE to prove every statement needed in order to make a mathematical argument. To not give any credit would be to say that all the hard work that the student has put in would be meaningless. This is counter-productive. The hard work and attempts should be worth something, as it is the hard work which will ensure success down the road. Besides, everything that was needed was there anyway.